1. Routing, Flow, and Capacity Design in Communication and Computer Networks Chapter 8: Fair Networks
Slides by
Yong Liu1, Deep Medhi2, and Michał Pióro3
1Polytechnic University, New York, USA
2University of Missouri-Kansas City, USA
3Warsaw University of Technology, Poland & Lund University, Sweden
October 2007

2. Outline Fair sharing of network resource Max-min Fairness
Proportional Fairness
Extension

3. Fair Networks Elastic Users:
demand volume NOT fixed
greedy users: use up resource if any, e.g. TCP
competition resolution?
Fairness: how to allocate available resource among network users.
capacitated design: resource=bandwidth
uncapacitated design: resource=budget
Applications
rate control,
bandwidth reservation
link dimensioning

4. Max-Min Fairness: definiation
Lexicographical Comparison
a n-vector x=(x1,x2, …,xn) sorted in non-decreasing order (x1≤x2 ≤ …≤ xn) is lexicographically greater than another n-vector y=(y1,y2, …,yn) sorted in non-decreasing order if an index k, 0 ≤k ≤n exists, such that xi =yi, for i=1,2,…,k-1 and xk >yk
(2,4,5) >L (2,3,100)
Max-min Fairness: an allocation is max-min fair if its lexicographically greater than any feasible allocation
Uniqueness?

5. Other Fairness Measures
Proportional fairness [Kelly, Maulloo & Tan, ’98]
A feasible rate vector x is proportionally fair if for every other feasible rate vector y
Proposed decentralized algorithm, proved properties
Generalized notions of fairness [Mo & Walrand, 2000]
-proportional fairness: A feasible rate vector x is
fair if for any other feasible rate vector y
Special cases: : proportional fairness
: max-min fairness

6. Capacitated Max-Min Flow Allocation
Fixed single path for each demand
Proposition: a flow allocation is max-min fair if for each demand d there exists at least one bottle-neck, and at least on one of its bottle-necks, demand d has the highest rate among all demands sharing that bottle-neck link.

7. Max-min Fairness Example
Session 3
Session 2
Session 1
C=1
C=1
Session 0
Max-min fair flow allocation
sessions 0,1,2: flow rate of 1/3
session 3: flow rate of 2/3

8. Max-Min Fairness: other definitions
Definition1: A feasible rate vector is max-min fair if no rate can be increased without decreasing some s.t.
Definition2: A feasible rate vector is an optimal solution to the MaxMin problem iff for every feasible rate vector with , for some user i, then there exists a user k such that and

9. How to Find Max-min Fair Allocation?
Idea: equal share as long as possible
Procedure
start with 0 rate for all demand
increase rate at the same speed for all demands, until some link saturated
remove saturated links, and demands using those links
go back to step 2 until no demand left.

10. Max-min Fair Algorithm

11. Max-min Fair Example link rate: AB=BC=1, CA=2 B demand 4 =2/3

12. Extended MMF
lower and upper bound on demands
weighted demand rate

13. Extended MMF: algorithm

14. Deal with Upper Bound
Add one auxiliary virtual link with link capacity wdHd for each demand with upper bound Hd

15. MMF with Flexible Paths
one demand can take multiple paths
max-min over aggregate rate for each demand
potentially more fair than single-path only
more difficult to solve

16. Uncapacitated Problem
Max-min fair sharing of budget
Formulation

17. Uncapacitated Problem
max-min allocation
all demands have the same rate
each demand takes the shortest path
proof?

18. Proportional Fairness
Proportional Fairness [Kelly, Maulloo & Tan, ’98]
A feasible rate vector x is proportionally fair if for every other feasible rate vector y
formulation

19. Linear Approximation of PF

20. Extended PF Formulation

21. Uncapacitated PF Design
maximize network revenue minus investment